Here are your correct matches, with a few bits of added info:
1. First president to visit Communist China = Richard Nixon
- This was a lasting foreign policy initiative, starting friendly relations between the US and The People's Republic of China
2. Egyptian leader = Anwar el-Sadat
3. Israeli leader = Menachem Begin
- Sadat and Begin signed a peace treaty between their countries.
4. deposed Shah of Iran = Mohammed Reza Pahlavi
5. Soviet leader visited by Nixon in 1972 = Leonid Brezhnev
6. Nixon's Vice President = Spiro Agnew
- Agnew was the Vice-President until he resigned for tax evasion. He was replaced by Gerald Ford
7. leader of the Islamic Republic of Iran = Ayatollah Khomeini
8. fired by President Nixon Archibald Cox
- Cox was Special Prosecutor in the Watergate investigation. Firing Cox did not solve that crisis for Nixon. Leon Jaworski was appointed to replace Cox, and the investigation continued.